Symplectic geometry has its origin in physics, but has flourished as an independent subject in mathematics, together with its offspring, symplectic

Transcription

1 Symplectic geometry has its origin in physics, but has flourished as an independent subject in mathematics, together with its offspring, symplectic topology. Symplectic methods have even been applied back to mathematical physics; for example, Floer theory has contributed new insights to quantum field theory. In a related direction, noncommutative geometry has developed an alternative mathematical quantization scheme, based on a geometric approach to operator algebras. Deformation quantization, a blend of symplectic methods and noncommutative geometry, approaches quantum mechanics from a more algebraic viewpoint, as it addresses quantization as a deformation of Poisson structures. This volume contains seven articles based on lectures given by invited speakers at two May 2010 workshops held at MSRI: Symplectic and Poisson Geometry in Interaction with Analysis, Algebra and Topology (honoring Alan Weinstein, one of the key figures in the field) and the Hayashibara Forum on Symplectic Geometry, Noncommutative Geometry and Physics. Both workshops were jointly sponsored by MSRI, the Research Institute of Mathematical Sciences at Kyoto University (RIMS), and the Hayashibara Foundation. The articles include presentations of previously unpublished results and comprehensive reviews including recent developments in these areas.

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3 Mathematical Sciences Research Institute Publications 62 Symplectic, Poisson, and Noncommutative Geometry

4 Mathematical Sciences Research Institute Publications 1 Freed/Uhlenbeck: Instantons and Four-Manifolds, second edition 2 Chern (ed.): Seminar on Nonlinear Partial Differential Equations 3 Lepowsky/Mandelstam/Singer (eds.): Vertex Operators in Mathematics and Physics 4 Kac (ed.): Infinite Dimensional Groups with Applications 5 Blackadar: K-Theory for Operator Algebras, second edition 6 Moore (ed.): Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics 7 Chorin/Majda (eds.): Wave Motion: Theory, Modelling, and Computation 8 Gersten (ed.): Essays in Group Theory 9 Moore/Schochet: Global Analysis on Foliated Spaces, second edition Drasin/Earle/Gehring/Kra/Marden (eds.): Holomorphic Functions and Moduli Ni/Peletier/Serrin (eds.): Nonlinear Diffusion Equations and Their Equilibrium States 14 Goodman/de la Harpe/Jones: Coxeter Graphs and Towers of Algebras 15 Hochster/Huneke/Sally (eds.): Commutative Algebra 16 Ihara/Ribet/Serre (eds.): Galois Groups over Q 17 Concus/Finn/Hoffman (eds.): Geometric Analysis and Computer Graphics 18 Bryant/Chern/Gardner/Goldschmidt/Griffiths: Exterior Differential Systems 19 Alperin (ed.): Arboreal Group Theory 20 Dazord/Weinstein (eds.): Symplectic Geometry, Groupoids, and Integrable Systems 21 Moschovakis (ed.): Logic from Computer Science 22 Ratiu (ed.): The Geometry of Hamiltonian Systems 23 Baumslag/Miller (eds.): Algorithms and Classification in Combinatorial Group Theory 24 Montgomery/Small (eds.): Noncommutative Rings 25 Akbulut/King: Topology of Real Algebraic Sets 26 Judah/Just/Woodin (eds.): Set Theory of the Continuum 27 Carlsson/Cohen/Hsiang/Jones (eds.): Algebraic Topology and Its Applications 28 Clemens/Kollár (eds.): Current Topics in Complex Algebraic Geometry 29 Nowakowski (ed.): Games of No Chance 30 Grove/Petersen (eds.): Comparison Geometry 31 Levy (ed.): Flavors of Geometry 32 Cecil/Chern (eds.): Tight and Taut Submanifolds 33 Axler/McCarthy/Sarason (eds.): Holomorphic Spaces 34 Ball/Milman (eds.): Convex Geometric Analysis 35 Levy (ed.): The Eightfold Way 36 Gavosto/Krantz/McCallum (eds.): Contemporary Issues in Mathematics Education 37 Schneider/Siu (eds.): Several Complex Variables 38 Billera/Björner/Green/Simion/Stanley (eds.): New Perspectives in Geometric Combinatorics 39 Haskell/Pillay/Steinhorn (eds.): Model Theory, Algebra, and Geometry 40 Bleher/Its (eds.): Random Matrix Models and Their Applications 41 Schneps (ed.): Galois Groups and Fundamental Groups 42 Nowakowski (ed.): More Games of No Chance 43 Montgomery/Schneider (eds.): New Directions in Hopf Algebras 44 Buhler/Stevenhagen (eds.): Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography 45 Jensen/Ledet/Yui: Generic Polynomials: Constructive Aspects of the Inverse Galois Problem 46 Rockmore/Healy (eds.): Modern Signal Processing 47 Uhlmann (ed.): Inside Out: Inverse Problems and Applications 48 Gross/Kotiuga: Electromagnetic Theory and Computation: A Topological Approach 49 Darmon/Zhang (eds.): Heegner Points and Rankin L-Series 50 Bao/Bryant/Chern/Shen (eds.): A Sampler of Riemann Finsler Geometry 51 Avramov/Green/Huneke/Smith/Sturmfels (eds.): Trends in Commutative Algebra 52 Goodman/Pach/Welzl (eds.): Combinatorial and Computational Geometry 53 Schoenfeld (ed.): Assessing Mathematical Proficiency 54 Hasselblatt (ed.): Dynamics, Ergodic Theory, and Geometry 55 Pinsky/Birnir (eds.): Probability, Geometry and Integrable Systems 56 Albert/Nowakowski (eds.): Games of No Chance 3 57 Kirsten/Williams (eds.): A Window into Zeta and Modular Physics 58 Friedman/Hunsicker/Libgober/Maxim (eds.): Topology of Stratified Spaces 59 Caporaso/M c Kernan/Mustață/Popa (eds.): Current Developments in Algebraic Geometry 60 Uhlmann (ed.): Inverse Problems and Applications: Inside Out II 61 Breuillard/Oh (eds.): Thin Groups and Superstrong Approximation

5 Symplectic, Poisson, and Noncommutative Geometry Edited by Tohru Eguchi Rikkyo University Yakov Eliashberg Stanford University Yoshiaki Maeda Keio University

6 Tohru Eguchi Rikkyo University Yoshiaki Maeda Keio University Yakov Eliashberg Stanford University Silvio Levy (Series Editor) Mathematical Sciences Research Institute The Mathematical Sciences Research Institute wishes to acknowledge support by the National Science Foundation and the Pacific Journal of Mathematics for the publication of this series. 32 Avenue of the Americas, New York, NY , USA Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. Information on this title: c Mathematical Sciences Research Institute 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United States of America A catalog record for this publication is available from the British Library. ISBN Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.